Invariants Under Transformation Math Example 3

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Example 3

medium
A figure is reflected over the xx-axis. Determine whether each property is invariant: (a) side lengths, (b) orientation (clockwise/counterclockwise), (c) area, (d) angle measures.

Solution

  1. 1
    Reflection is an isometry, so distances are preserved: (a) side lengths are invariant, (c) area is invariant, (d) angle measures are invariant.
  2. 2
    However, reflection reverses orientation: a clockwise-labeled figure becomes counterclockwise. So (b) orientation is NOT invariant.

Answer

(a)Ā invariant,Ā (b)Ā notĀ invariant,Ā (c)Ā invariant,Ā (d)Ā invariant\text{(a) invariant, (b) not invariant, (c) invariant, (d) invariant}
Reflections preserve distances, angles, and area (they are isometries), but they reverse the orientation of the figure. This is why a reflected image is a mirror image ā€” congruent but not directly superimposable without flipping.

About Invariants Under Transformation

A property of a function is invariant under a transformation if it remains unchanged after the transformation is applied to the function.

Learn more about Invariants Under Transformation ā†’

More Invariants Under Transformation Examples

Example 1 easy

A triangle with vertices at [formula], [formula], and [formula] is translated by the vector [formula

Example 2 medium

A rectangle with vertices [formula], [formula], [formula], [formula] is dilated by a scale factor of

Example 4 hard

Under a shear transformation defined by [formula], determine whether the area of a unit square with

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